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A Story of Logic
These brief notes on the history of logic are meant to be an easily understood basis for
studying the great ideas in logic. Appreciating how the basic concepts were discovered
and why they were needed is an accessible basis for a technical understanding of these
concepts For many former students, this is a route by which they have come to deeply
understand and appreciate logic. Logic is one of the oldest continuously studied academic
subjects, and while investigating its central questions, Alan M. Turing gave birth to
computer science in 1936 as you will see toward the end of these notes. He and his thesis
advisor, Alonzo Church, laid theoretical foundations for computer science that continue
to frame some of the deepest questions and provide the conceptual tools for important
applications. Indeed, their ideas underlie one of the most enduring contributions of
computer science to intellectual history, the discovery that we can program computers to
assist us in solving conceptual problems that we could not solve without them. It is no
longer in doubt that computers can be programmed to perform high level mental
operations that were once regarded as the pinnacle of human mental activity.
Let us start as far back as is historically sensible as far as we know.
Epimenides of Crete circa 500 BCE
Epimenides is reputed to have posed a version of the Liar’s Paradox, “All Cretans are
liars” said a known Cretan. This paradox is mentioned in the Bible (Paul’s epistle to
Titus). A more modern related paradox asks about the truth or falsity of the statement:
This sentence is false.
Clearly this apparently declarative sentence cannot have a truth value as normally
understood, i.e. to be either true or false. To this day logicians do not have a definitive
accounting of the meaning of this paradoxical statement and others like it. In the precise
logical languages we study in this course, no such sentence is allowed to be “well
formed.” This suggests that natural language cannot be treated as a precise logical
language.
Aristotle
Aristotle wrote one of the most influential books on logic, called by his followers the
Organon, circa 330 BCE (the date is my approximation of the date when it might have
been written, probably at the Lyceum academy in Athens during the time of Ptolemy I).
Up until the time of Kant it was understood to be a definitive treatment of logic – for
about two thousand one hundred years (2,100 years).
Aristotle said: “Deduction is speech (logos) in which certain things having been
supposed, something different from these supposed results follows of necessity from their
being so.” (Prior Analytics I.2, 24b 18-20.)
See http://plato.stanford.edu/entries/aristotle-logic/
Speech consists of sentences, each has the form of a subject and a predicate. We continue
to this day to discuss grammar in these terms. When we say Socrates is mortal, the
subject is Socrates and the predicate is is-mortal. For Aristotle, subjects and predicates
are terms in his logic, either individual terms like Socrates or universal terms such as is
mortal.
Aristotle’s logic is about deduction from premises to conclusion. The notion that the
conclusion “follows of necessity” is the modern idea of logical consequence which we
study in this course. His deductions are presented in a very specific form called a
syllogism. Here is a famous example.
All Men are Mortal
Socrates is a Man
_______________
Socrates is Mortal
premise one
premise two
conclusion
Aristotle worked out a detailed account of syllogisms including rules for creating an
argument from a sequence of them. There are good examples in a systematic modern
notation available on-line, as in http://plato.stanford.edu/entries/aristotle-logic/ There is
something very modern in his approach in that he proves properties of his logical system
using a more informal style. We call such a study of a logical system, metalogic.
Aristotle used his logic for deducing scientific results, and he famously said that
1. Whatever is scientifically known must be demonstrated.
2. The premises of a demonstration must be scientifically known.
This account allowed “agnostics” to argue that scientific knowledge is impossible
because the premises must be demonstrated as well, so there is an infinite regress in
trying to find even one statement which is scientifically known. Aristotle claimed that the
demonstration process ends when we reach certain premises which we accept based on a
state of mind called nous (intuition, insight, intelligence). He claimed that the human
mind was capable of certain intuitions needed for science. We will see this idea emerge
with great force in the philosophy of mathematics and to have a significant impact on the
practice of computer science.
Euclid
Euclid’s Elements (circa 306 BC) are not always considered in the history of logic, but
they are critical in understanding the use of logic in mathematics and computer science. I
trust that readers will be familiar with Euclidean geometry as presented in secondary
school. As with Euclid, school accounts begin with definitions of the objects such as
points, lines, line segments, straight lines, figures, circles, plane angles, and so forth for
23 definitions in the Elements. We can think of these definitions as also establishing
types or classes, e.g. the type of Points, the type of Figures, Circles (a subtype of
Figures), and so forth. Programmers use such types or classes when they write geometric
algorithms.
Euclid stated five basic axioms (postulates) about geometry, but the first three don’t look
like the subject predicate declarative sentences of Aristotle. Here are simple versions of
the first four axioms.
1.
2.
3.
4.
To draw a straight line from any point to any point.
To extend a line to a straight line.
To draw a circle with given center and radius.
All right angles are equal.
The first three axioms are statements about what can be constructed. Euclid is saying that
it is possible to perform certain basic constructions, and to understand the axioms is to
know how to perform the constructions.
Euclid’s famous fifth axiom is paraphrased this way.
5. If a straight line falling on two straight lines makes interior angles on the same
side less than two right angles, then the two straight lines meet at a point on the
side where the angles are less than right angles.
Ever since Descartes introduced analytic geometry and translated geometric questions
into analysis, mathematicians have realized that there are other rigorous ways to study
geometry beyond Euclid's. Nevertheless Euclidean geometry has been the subject of
sustained axiomatic and logical analysis right up to the present. New proofs
in the style of Euclid are being published even now such as the Steiner-Lehmus theorem
that if two angle bisectors of a triangle are equal in length, then the triangles must be
isosceles see [GM63]. Two of the most cited works are Hilbert's book Foundations of
Geometry and Tarski's decision procedure for geometry based on the first-order theory of
real closed fields, see [TG99] and von Plato's axiomatization in type theory [von95].
Steiner-Lehmus theorem:
If angle bisectors are equal, then
the triangle is isosceles.
Leibniz
The mathematician Leibniz co-invented the calculus with Newton, and he also introduced
a conceptual advance in our understanding of the scope and potential of logic, a
conception going beyond that of Aristotle in some important ways. He believed that logic
was a science that contained concepts essential to all other sciences.
Leibniz also is a major node in the genealogical tree of mathematicians and logicians.
Many logicians since the 1660’s can trace their academic ancestry back to Leibniz,
including me.
Here is a quotation by Leibniz from his De Arte Combinatoria, 1666. It is taken from an
article by J. Bates, T. Knoblock and me entitled Writing Programs that Construct Proofs,
Journal of Automated Reasoning, vol. 1, no. 3, pp. 285-326, 1984.
Boole
Boole wrote The Laws of Thought in 1854. On page 27 of the Dover 1958 reprinting we
see the propositional calculus similar to how it is taught today but using the symbols from
algebra, +, -, x, and =. The sign + corresponds to what we now call exclusive or. In this
major advance over Aristotle, Boole studies formulas built out of logical operators whose
meaning is defined in terms of the truth values, say 0 and 1 for false and true. Boole’s
presentation is inspired by the laws of algebra, and he hopes to present methods for
solving logical equations. The propositional calculus is also called Boolean algebra when
presented in Boole’s original style. The laws of this algebra are especially important in
analyzing binary computer circuits (with values 0 and 1). In honor of Boole we also call
the truth values Booleans and denote them by B.
Frege
The most significant achievement in logic after Aristotle is Frege’s publication in 1879 of
his Begriffsschrift (concept-script, concept-writing, idea-writing, ideography). It is a
booklet of 88 pages that changed logic forever. http://plato.stanford.edu/entries/frege/
In this work he invented what we now call first-order logic, a major topic of this course.
He started by making a case for advancing Leibniz’s conception of logic and arguing that
the Aristotelian conception was too narrowly tied to natural language. He said: “These
derivations (in notation) from what is traditional find their justification in the fact that
logic has hitherto always followed ordinary language and grammar too closely. In
particular, I believe that the replacement of the concepts subject and predicate by
argument and function, respectively, will stand the test of time.”
Frege was attempting to understand the notion of a sequence very precisely. The gradual
arithmetization of analysis and the calculus (see below), had reduced the idea of a real
number to the notion of sequences of rational numbers, and rational numbers could be
reduced to pairs of integers, and integers could be reduced to natural numbers. This
reduction focused attention on defining natural numbers and the notion of a sequence.
There were questions about how to prove properties of sequences.
“The most reliable way of carrying out a proof, obviously, is to follow pure logic, a way
that, disregarding the particular characteristics of objects, depends solely on those laws
upon which all knowledge rests.” He goes on to say how hard it is to do this using
ordinary language. “To prevent anything intuitive from penetrating here unnoticed, I had
to bend every effort to keep the chain of inferences free of gaps. In attempting to comply
with this requirement in the strictest possible way I found the inadequacy of language to
be an obstacle. … This deficiency led me to the idea of the present ideography.”
Here is how he elaborates on Leibniz: “Leibniz, too, recognized – and perhaps overrated
– the advantages of an adequate system of notation. His idea of a universal
characteristic, of a calculus philosophicus or ratiocinator, was go gigantic that the
attempt to realize it could not go beyond the bare preliminaries. The enthusiasm that
seized its originator when he contemplated the immense increase in the intellectual
power of mankind that a system of notation directly appropriate to objects themselves
would bring about led him to underestimate the difficulties that stand in the way of such
an enterprise. But, even if this worthy goal cannot be reached in one leap, we need not
despair of a slow, step-by-step approximation. When a problem appears to be unsolvable
in its full generality, one should temporarily restrict it; perhaps in can then be conquered
by a gradual advance.”
Below are two pages showing the unusual notation in Begriffsschrift along with Frege’s
explanation of the problem he was formalizing. This notation is no longer used, but it has
distinct advantages in seeing the binding structure of formulas, a topic we will discuss at
length in this course.
One of the important innovations in Begriffsschrift is the notion of judgements. Frege
distinguished between judging that a formula is true and indicating that it has content and
could be judged. The content indicator is written – A for a purported formula A and the
truth judgement is written with the turnstile |- to claim that A is true or at least worth
trying to prove. We see |- A in the above document (we sometimes write ─A and ├A
respectively for A having propositional content and A being judged for truth, thus
wanting proof).
The Nineteenth Century “Crisis” in Analysis
During the 1800’s analysis (“very advanced calculus”) rapidly developed, despite the fact
that many of the basic concepts such as function, sequence, set, continuity, convergence,
infinite series, infinitesimal, etc. were not clear. For example, Cauchy believed that every
continuous function was differentiable and that the infinite sum of a sequence of
continuous functions was continuous –both of which are false; and yet he was the person
who gave a precise definition of continuity and was one of the leading figures in analysis.
Many mathematicians believed that an infinite series made sense as an object even if it
did not converge, but they were not sure what they meant. Even Euler believed that every
infinite series was the result of expanding a unique finite formula. Gradually these
concepts were clarified by many gifted mathematicians such as Abel, Bolzano, Borel,
Bernoulli, Dedekind, Dirichlet, Fourier, Heine, Jordan, Lebesgue, Poincare, Riemann,
Weierstrass, and others for whom many of the theorems of analysis are named.
The Arithmetization of Analysis
During this period there were significant advances in understanding the concept of a real
number, the very foundation of analysis. One approach to real numbers since Descartes
was to ground them in geometry and consider real numbers as points on the line. But with
the discovery of non-Euclidean geometries, people began to think that geometric “truths”
were not absolute and perhaps not a secure basis for understanding the reals. Gauss
expressed the opinion that only the natural numbers were a source of absolute
mathematical truth.
Cauchy was able to reduce the concept of a real number to that of a sequence of rational
numbers that converged, so called Cauchy sequences. Thus he reduced the idea of real
number to the idea of functions and rational numbers. Rational numbers could be
understood as pairs of integers, and integers could be explained as natural numbers. So
the ultimate foundation for mathematics depended on understanding functions and
natural numbers.
Peano
In 1889 Peano published his small book The Principles of Arithmetic in which he showed
how to use symbolic logic to axiomatize a theory of natural numbers, now known as
Peano Arithmetic. He used Boole’s ideas for propositional logic and Schröder’s notation
for quantifiers such as “for all” and “there exists.”
Cantor
By 1899 Cantor knew there was a paradox in his conception of set theory under the
assumption that there is a set M of all sets (Menge is the German word for set). Cantor
had defined the cardinality of any set A, denoted |A|. He proved his famous theorem that
the cardinality of the set of all subsets of any set A, call it Pow(A), has a larger
cardinality than A, that is |Pow(A)| > |A|. But if M is the set of all sets, then Pow(M) is a
subset of M, so |Pow(M)| < |M|, which contradicts his theorem. His conclusion is that
there can’t be a set M of all sets, such a set is in some ways “too big.” This is Cantor’s
Paradox.
Russell
In 1902, Russell discovered a paradox along similar lines as Cantor’s, but much simpler
to explain. He considered the set of all sets that do not contain themselves, call it R and
define it as R = {x | not(x ε x)}. Now Russell asks whether R ε R. We can see
immediately that if R ε R, then by definition of R, we know not(R ε R). If not(R ε R),
then by definition of R, R ε R. Since according to classical logic either R belongs to R or
it does not, and since each possibility leads to a contradiction, we have a contradiction in
classical mathematics if we assume there is such a set as R. This is known as Russell’s
Paradox.
In trying to sort out the reasons behind this paradox, Russell was led to formulate his
famous Theory of Types in the 1908 book Mathematical Logic Based on a Theory of
Types. Then in the period from 1910 to 1925 Whitehead and Russell wrote and published
a three volume book based on this type theory and designed to be a secure logical
foundation for all of mathematics entitled Principia Mathematica (PM). We will
examine aspects of this type theory during the course, and we will use type theory from
the beginning in our informal mathematics because the notion of type is now central in
computer science, owing largely to early research in Britain, France, Sweden and the US
as well as to the increasing role of type systems in the design of programming languages.
Zermelo
Meanwhile in 1908 the German mathematician Zermelo published an influential paper
Investigations in the foundations of set theory in which he organized a set theory, now
called Z, around a collection of methods for forming sets that he thought were safe from
the paradoxes of Cantor, Russell and others. He did not allow the set of all sets to be a
set, and he restricted the methods of building sets starting from a single infinite set used
to build the natural numbers (axiom of infinity). Other sets could be built using the
separation axiom: if S is a set and P a definite property on sets, then {x:S | P(x)} is a set.
In 1922 this notion of “definite property” was made more precise by the logicians
Fraenkel and Skolem, and by Weyl. It is now standard practice to require that P be a
formula of first- order logic using the primitive concept of set membership normally
denoted by epsilon, ε. Fraenkel and Skolem also suggested a powerful new axiom
deemed by many to be safe, called the axiom of replacement. This set theory is now
called ZF set theory and is considered by many mathematicians to be the “standard
foundation” for modern mathematics; however, it is not able to express ideas from
category theory and type theory without extension.
Knonecker
Kronecker was a distinguished German mathematician known for his work in algebraic
and analytic number theory. He is also known for being opposed to using the concept of a
completed infinite set in mathematical reasoning as Cantor and others were starting to do.
He agreed with Gauss that the idea of an infinite set was just a manner of speaking about
collections such as the natural numbers, 0,1,2,3, … which can be continued without end
because given any number n, we can construct a larger number by adding one to it. He
was in particular opposed to Cantor’s work on set theory, but he also disagreed with
methods of proof that did not produce concrete answers. He strongly favored
computational methods and explicit constructions.
Kronecker is mentioned in Bell’s Men of Mathematics as being “viciously opposed” to
Cantor and others who proved results about infinite sets as if they were completed
totalities, but this appears to be a considerable exaggeration. What is true is that in 1886
he made an after dinner speech in which he said “God made the integers, all else is the
work of man” as a way of expressing his interest in constructions. Here is the German for
what he said:
"Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk".
We will look next at a mathematician who did openly attack non-computational methods
and who was in contention for being the leading mathematician of his day.
Brouwer
In his 1907 doctoral dissertation, On the Foundations of Mathematics, Brouwer provided
a meaning for mathematical statements based on mental constructions. These
constructions are intuitively known to be effective for basic mathematical tasks. This
philosophical stance is known as intuitionism, and Brouwer was interested in building
mathematics according to this philosophy. Brouwer believed that the crisis in the
foundations of analysis was due to mathematicians not understanding the full extent of
constructive methods. In particular he believed that our mental constructions are the
proper justification for logic and that they do not justify the full logic of Aristotle which
is based on affirmation and denial of sentences. Restating this in terms of propositional
or Boolean logic, one of the fundamental laws that is not justified according to Brouwer
is the law of excluded middle, P or not P, for any proposition P. For Brouwer, to assert P
is to know how to prove it, and he could imagine propositions which we could never
prove or disprove.
Brouwer believed that logic was the study of a particular subset of abstract constructions
but that it played no special role in the foundations of mathematics, it was simply a form
of mathematics.
Brouwer believed that our mathematical intuitions concern two basic aspects of
mathematics, the discrete and the continuous. Our intuitions about discreteness and
counting discrete objects give rise to number theory, and our intuitions about time give
rise to the notion of continuity and real numbers. He believed that mathematicians had
not recognized the rich constructions need for understanding real numbers, including for
computing with real numbers. He was determined to study these constructions and thus
settle the disputed issues in analysis and the theory or real numbers. But first he decided
he should establish himself as “the best mathematician in the world.” He proceeded to
develop point set topology and proved his famous and widely used fixed point theorem.
He attracted several concerts and fellow travelers, and when one of the most promising
young mathematicians, Weyl, became a follower of Brouwer, another contender for
“world’s best mathematician, Hilbert, entered the fray in a “frog and mouse war” with
Brouwer. He is famous for saying that “we can know and must know” the truth or falsity
of any mathematical statement.
Heyting
Brouwer was dismissive of formal logic, so he did not write down a logic that could be
used for intuitionistic reasoning. However in 1930 Heyting wrote The formal rules of
intuitionistic logic. This was the creation of another branch of logical systems that
respected intuitionistic principles. Heyting also formalized a theory of arithmetic similar
to Peano Arithmetic (PA) but using intuitionistic logic; it is called Heyting Arithmetic
(HA). We will study these logics in this course and relate them. In 1925 Kolmogorov
wrote a precursor work entitled On the principle of excluded middle, in which he
axiomatized a fragment of intuitionistic logic.
A key feature of these new accounts of logic is that the notion of truth is replaced by the
notion of knowledge. The logic is centered on explaining how we can know propositions.
This semantics is known as BHK semantics and we will study it in depth. It leads to the
major new principle of logic called the propositions-as-types principle which we will
also study in depth.
Hilbert
Hilbert had been thinking about the foundations of mathematics since 1899 when he
wrote The Foundations of Geometry. In 1904 he began thinking more broadly about the
foundations of logic and arithmetic, but it was not until 1925 that he began serious work
on these topics, sketched in his article On the Infinite. He responded to Principia
Mathematica by showing how to create a completely formal logical theory. Nowadays
we mean by this a theory that can be implemented in software, but Hilbert captured this
concept before we knew about computers. He treated a formal theory as the basis for a
meaningless game played according to strict rules; the goal of the game was to prove
theorems. The important property of the game was that it was consistent; that is, it would
not be possible to prove a theorem P and also prove not P.
Hilbert was willing to admit that the idea of actual infinity might be meaningless, merely
a convenient way of thinking, an idealization. He imagined that by his method of
formalizing theories and analyzing them in a constructive metatheory, he could justify the
most free-ranging imagination, the creation of ideal worlds in which there were actual
infinities. He called such a world a paradise and said that people like Brouwer could not
“drive us out of this paradise” because we could use strict methods of proof, constructive
methods, to show that the game of set theory or the game of type theory was consistent.
He even believed that the game could settle all questions; it could be complete in the
sense that all true theorems of the theory could be proved. Hilbert’s program for saving
mathematics was bold and inspiring. His program was also a way to save mathematics as
it was being practiced. He saw himself as a savior of mathematics from critics like
Kronecker, Brouwer, and his brilliant former student Weyl.
In the period from 1925 until 1940 mathematics was at a tipping point. It could go with
Brouwer and his many followers and become constructive mathematics or it could remain
the classical mathematics that Hilbert treasured and advocated, open to very nonconstructive methods of the kind Hilbert had used in his ascendancy into the group of the
very best mathematicians of his day, a group that included at least Brouwer and Poincare
as well.
Gödel
In 1931 Gödel published his article On formally undecidable propositions of Principia
Mathematica and related systems. He proved that Principia Mathematica (PM) could not
be complete if it was consistent; that is, unless PM could prove everything, there was a
true sentence of that theory which was not provable. This surprising discovery seemed to
put an end to Hilbert’s program for saving mathematics. It was a stunning result. It also
showed that perhaps Brouwer was right and there would be statements in mathematics
that we could never prove even though they might be true in some sense, say in the sense
that we could never disprove them either.
Church
By 1940 Gödel had moved from Austria to the Institute for Advanced Study in Princeton.
Alonzo Church was a professor of mathematics at Princeton University where for the
year 1937 Alan Turing had also come to study logic with Church. In 1936 Church
published An unsolvable problem of elementary number theory and the short note A note
on the Entscheidungsproblem. The Entscheidungsproblem, or decision problem, was one
of the famous open problems that Hilbert had posed in 1928. Turing had also solved this
problem as we will discuss below. http://en.wikipedia.org/wiki/Hilbert%27s_problems
Church’s solution was in terms of his lambda calculus, a formalism later shown to be
equivalent to Turing machines and all other known formalisms for expressing the
computable functions. In a sense the lambda calculus was the first programming
language, and indeed it led John McCarthy and his students to create the programming
language Lisp circa 1960.
Church was the doctoral thesis advisor for a very large number of logicians. According to
the Mathematics Genealogy Project http://genealogy.math.ndsu.nodak.edu/ Church has
produced 2,697 descendants, a number that keeps growing. Among them are some of the
most famous logicians in the world such as Alan Turing (1938), Dana Scott (1958), a
Turing Award winner, Michael Rabin (1957) a Turing Award winner, Simon Kochen
(1959), Stephen Kleene (1934), J. B. Rosser (1934), Martin Davis (1950), and Raymond
Smullyan (1959), the author of our textbook.
Church is also known for developing the Simple Theory of Types which is the logical
system implemented in the HOL theorem prover and widely used in computer science
applications, starting with hardware verification.
Turing
In 1936 Turing published his ground breaking paper On computable numbers with an
application to the Entscheidungsproblem. He discovered this result before he came to
Princeton to work with Church. In this paper he defines computable real numbers using
what has become known as Turing Machines. Turing’s model of computation is still
widely taught, and his unsolvable problem is taught to computer science freshmen around
the world, the halting problem. Turing showed that no Turing Machine can solve the
problem of whether a given such machine will halt on a given input or “on blank tape.”
This model of computation is very natural and convincing as a general model of how
people and machines compute in principle. Gödel was extremely impressed by this
model, even though he imagined a competing idea, the so called Herbrand-Gödel
equations for general recursive functions. He said that Turing’s model showed that the
idea of computability was absolute, that is independent of any particular logic. At the
same time, Gödel did not appreciate Church’s lambda calculus which also defined the
class of all computable functions. Indeed the famous Church-Turing Thesis is the claim
that these formalisms define what mathematicians mean by effectively computable
functions. The difference is that Turing machines are a simple machine model and the
lambda calculus is a very high-level functional programming language. Church put to
Turing the task of proving that the lambda calculus could be translated into Turing
Machines. Thus Turing was assigned the task of building the first compiler for a high
level language. Perhaps from that moment on he became a computer scientist. In any
case, he went on to contribute broadly to what we now consider to be computer science.
During World War II, Turing worked at Bletchley Park in England as a code breaker. His
role during the war was secret until the 1970’s. Since then much has been written about
his heroic role. One of the most accessible books is The Man Who Knew Too Much by
David Leavitt.
Bishop
In 1967 Bishop published his extremely influential book on constructive mathematics
called Foundations of Constructive Analysis. In this book he showed that it is possible to
develop most of modern analysis without adopting ideas from Brouwer’s intuitionism
that contradict classical mathematics. For example, for in Brouwer’s analysis, all
functions from the reals to the reals are continuous. Bishop’s analysis is entirely
consistent with classical analysis, yet all the functions are computable and all the results
are proved constructively. This remarkable achievement seems to me a good example of
American ingenuity. Bishop enjoys all the positive benefits of intuitionism without
attacking classical mathematics. It is a philosophy of co-existence.
Martin-Löf
We now turn to the work of a living logician, Per Martin-Löf who created Intuitionistic
Type Theory (ITT) in 1972 and wrote about its connection to computer science in the
1982 paper Constructive Mathematics and Computer Programming. Per is a colleague
who has influenced my work, so I might be biased in my assessment. I will talk about his
theory ITT in this course and its impact on Computational Type Theory (CTT) which was
developed at Cornell as an extension of ITT more suitable as a foundation for computing.
The goal of Per’s work was to provide a firm logical foundation to Bishop’s analysis.
Later as he came to know more computer scientist, he broadened his goals to providing a
semantics for programming languages and logics.
Hoare
Another living figure who contributed to the logic of programming was the computer
scientist Tony Hoare who in 1969 published his paper An Axiomatic Basis for Computer
Programming. This paper created what is now called Hoare Logic, a logic for reasoning
about standard programming constructs such as assignment statements, while loops, and
recursive procedures. He also wrote a seminal paper on data types that could be
considered the first applied piece of type theory and help promote the idea that types are a
major organizing concept in programming languages and programming methodology.
Hoare had to face the fact that types in programming languages do not mean quite the
same thing as they do in mathematics. The functions of type nat to nat in a programming
language are partial functions; they may not terminate. In mathematics the functions
from N to N all terminate.
Milner
Last year at this time, September 2009, Robin Milner was also alive and was one of my
close colleagues. He is the inventor of the ML programming language and its type
inference algorithm. He published the very influential book Edinburgh LCF: A
Mechanized Logic of Computation with his colleagues Michael Gordon and Christopher
Wadsworth. The theory LCF is a formalization of Dana Scott’s logic of partial functions.
A long lasting contribution of this book is that most of the major interactive theorem
provers in use today (Coq, HOL, HOL-Light, Isabelle, Nuprl, and MetaPRL) all use the
LCF architecture and Milner’s idea of tactics.
Cornell PRL Group
Since 1984 this Cornell research group has been a leader in the subject of Computational
Type Theory and its automation. In 1986 we published the book Implementing
Mathematics with the Nuprl Proof Development System which we will mention in the
course. The Nuprl prover described in that book (Nuprl-3) is still actively used in
program verification and in formal methods research (Nuprl-5), especially software
security and protocol verification. In its current form it validates Brouwer’s notion
that the space of constructions (computations) is extraordinarily rich and extends
beyond the normal computable functions to include concurrent processes and other
forms of interaction. It even seems likely that we can justify Brouwer’s idea of free
choice sequences in applications.
http://www.scholarpedia.org/article/Computational_type_theory
References
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